1. Field of the Invention
The present invention is directed to a method for image reconstruction in imaging technology for the implementation of a fast convolution with a transformation length M while allowing slight over-convolution errors, whereby measured projections of a length N are convoluted with a modified filter kernel by means of Fast Fourier Transform (FFT) and Inverse Fast Fourier Transform (IFFT).
2. Description of the Prior Art
In imaging technology, for example computed tomography, data comprising projections of a patient obtained, for example, by a radiological measuring system rotating around the patient, are subjected to several mathematical operations for image reconstruction, including, among other things, convolution with a filter kernel. Extensive smearing of individual subject details in the reconstructed image, which would otherwise occur given an immediate back-projection of measured attenuation profiles (projections), are avoided by the convolution, which essentially corresponds to a high-pass filtering. The convolution is thus an important prerequisite for the determination of an immediate image, i.e., making the calculated image available immediately after the end of the measuring event. Since the reconstruction of images, consequently, must be undertaken rapidly, a technique known as "fast convolution", which is usually calculated cyclically for discrete functions, is utilized for the calculation of the convolution, i.e. for fast implementation of the convolution in a computer.
Gonzalez, R. C.; Woods, R. E., "Digital Image Processing", Addison-Wesley, June 1992, describes the convolution of two functions with Fast Fourier and Inverse Fast Fourier Transformation.
Vassiliadis, K. P., et al., "Reconstruction of Magnetic Resonance Images Using One-Dimensional Techniques", IEEE Transactions on Medical Imaging, Vol. 12, No. 4, December 1993, pp. 758-763, discloses the equivalency between a multi-dimensional and a one-dimensional discrete function.
When a vector formed from a projection is N elements long and is to be convoluted with a filter kernel, then the transformation length M must be M.gtoreq.2N -1 according to the theory of fast convolution in order to obtain error-free results from the fast convolution. Since only powers of 2 with the transformation length M=2.sup.m are usually available in the implementation of the fast convolution using Fast Fourier Transformation and Inverse Fast Fourier Transformation from vector libraries, the smallest power of 2 that meets the condition M=2.sup.m .gtoreq.2N -1 is consequently selected as the transformation length.
The selection of M is simple when N is a power of 2 such as 2.sup.m-1. When, however, the length N of the vector to be convoluted is not a power of 2 (for example, N=1536), then the transformation length M must be rounded up to the next power of 2 (for example, 4096 here). An unnecessary excess of M-(2N -1) values (for example, 1025 here) thereby arises that must be processed in the fast convolution, and can lead to an undesirably long calculating time of the fast convolution.
In order to reduce this increase in calculating time, a known method for fast "convolution with the next power of 2" shown in FIG. 1 can be employed when over-convolution errors are permitted. A vector formed from a projection with the length N (2.sup.m-1 .ltoreq.N&lt;2.sup.m, for example N.apprxeq.0.6*2.sup.m) is thereby expanded to the transformation length M=N+S=2.sup.m by attaching S zeros to the projection with the length N. The filter core h(k) in the spatial domain (=impulse response) employed for the fast convolution is calculated for a transformation length of M, whereby the impulse response can also be selected a few values shorter than M for image improvement. After the fast convolution with the transformation length M of the vector with the filter kernel h(k) selected somewhat too short, the S+1 middle values in the filtered vector having the length N are correct, whereas a total of N-S-1 values are falsified at the right and left edges of the projection. On the one hand, an over-convolution error thereby arises due to the overly short transformation length M of the filter kernel h(k), and a part of the projection having the length N is weighted with incorrect filter coefficients. Since, however, these incorrect filter coefficients lie at the outer edge of the filter core and are thus very small, and, since the test subject also exhibits lower attenuation values at the edge, the error is negligible given an adequately large number of S zeros attached to a projection having the length N.
When the method for fast "convolution with the next power of 2" is implemented on a computer, then there are two possibilities for the implementation of the convolution. First, transformation of a projection with the transformation length M/2 with post-processing that corrects the incomplete transformation, and, second, transformation of two projections, one in the real and one in the imaginary part of a transformation having the transformation length M.
When, however, the number S (for example S&lt;0.5*N) of zeros attached to the projection having the length N is too low, the above-described method leads to an excessively large error at the outside edge of the reconstructed image that can no longer be tolerated.